The present invention relates to a new and distinct blackberry cultivar designated ‘DrisBlackThirteen’ and botanically known as Rubus L. subgenus Rubus. This new blackberry variety was identified in Santa Cruz County, Calif. in December 2009 and originated from a cross between the proprietary female parent blackberry plant ‘BP571 (259L4)’ (unpatented) and the proprietary male parent blackberry plant ‘BP554 (252I5)’ (unpatented). The original seedling of the new cultivar was first asexually propagated by tissue culture and root cuttings at a nursery in Santa Cruz County, Calif. in 2009. ‘DrisBlackThirteen’ was subsequently asexually propagated by tissue culture and root cuttings and underwent further testing at a nursery in Santa Cruz County, Calif. from 2012 to 2014. The present invention has been found to be stable and reproduce true to type through successive asexual propagations.